Let's say we have two data sets, X containing values of years and Y containing values of prices of some item for those years. The prices follow an exponential $2^x$ pattern from year to year. Therefore we decide to log base 2 the Y data in order to have a better linear regression model.
My question is, now that we've logged our Y data (but not the X data), will our prediction model output logged values that will have to be "de-logged" with $2^y$ in order to be interpreted easily?
Whether it's true or not, I would appreciate some intuition as to why it's the case; for some reason I'm having trouble tracing exactly why it should be one way or the other, even though my intuition says we probably get logged predictions that we need to delog.
EDIT: I think in general I have a bit of a problem with transformations. Because I can't trace exactly what's happening, it never feels like a truly legitimate technique, kind of like cheating, the way proof by contradiction feels to a lot of people the first few times they're exposed to it.
